Question

3. A hive of bees contains 27 bees when it is first discovered. After 3 days, there are 36 bees. It is determined that the population of bees increases exponentially.

How many bees are will there be after 30 days? Express your answer as an exact value and rounded to the nearest whole number.

Answer 1

The population of bees in the hive increases exponentially. To find the number of bees after 30 days, we need to use the formula P = P0 * ekt, where P is the final population, P0 is the initial population, e is Euler's number, k is the growth rate, and t is the time in days. After performing the calculations, the approximate number of bees after 30 days is 110.

Step-by-step explanation:

The population of bees in the hive increases exponentially. To find the number of bees after 30 days, we can use the formula P = P0 * ekt, where P is the final population, P0 is the initial population, e is Euler's number (approximately 2.71828), k is the growth rate, and t is the time in days.

Given that the initial population is 27 bees and the population after 3 days is 36 bees, we can plug in these values to find the growth rate (k):

36 = 27 * e3k

Dividing both sides by 27, we get:

e3k = 36/27

e3k = 4/3

Taking the natural logarithm of both sides, we get:

3k = ln(4/3)

k = ln(4/3) / 3

Now we can find the population after 30 days:

P = 27 * e(ln(4/3) / 3) * 30

Rounding to the nearest whole number, there will be approximately 110 bees after 30 days.

Answer 2

P = 36^10 / 3^27

= 479 to the nearest whole number.

Step-by-step explanation:

Exponential increase;

36 = 27(c)^3

c^3 = 36/27

c = ∛36/3

So we have the expression:

P = 27(∛36/3)^t where t is the number of days and P = the population.

After 30 days we have

P = 27(∛36/3)^30

P = 27 * 36^10 / 3 / 3^30

P = 27* (36^1/3)^30 / 3^30

P = 3^3 * 36^10 / 3^30

P = 36^10 / 3^27.

P = 479.

P = 3^3 * 36^10 / 3 ^30

= 36^10 / 3^27